205 



(liv'isofs 34, 19, 10, 7, 10, give us 3y + 7, if such is a true 

 divisor, the co-factors 5, — 2, — 1, 2, 1, should supply (as 

 they do) a trinomial whose highest co-efficient is unit. 



But although the second series 10^ 1, —2, 1, 10, would 



apparently give the di\i3or 3y — Sjz+l, yet the co-factors 

 17, — 38, 5, 14, 1, will give us no cubical divisor, those co- 

 factors not havino; the third differences common : therefore 

 we reject the series 10, 1, — 3, 1, 10, as casual, and not 

 arising from the nature of the operation. 



To reject those numbers 10, 1, — 2, I, 10, Newton con- 

 tinues that series, according to the required law, by continuing 

 the terms of the arithmetical series of the remainders. Now, 

 of the scries 10, 1, — 2, 1, 10, 25, the new term 25, is found 

 not to divide — 19O, which results from the substitution of 

 — 2. But the method which is above given detects the for- 

 tuitous divisors, without the trouble of continuing them, 

 which is a particular advantage, if the fortuitous series should 

 not break off for the several terms. The co-factors are im- 

 mediately pointed out by the numeral divisors to be tried, 

 and besides the greater facility of numerical subduction than 

 of the division of algebraical quantities, it is also advan- 

 tageous that the above can be tried when the numerical di- 

 visors are first discovered to have a constant order of dif- 

 ferences, but algebraical division can only be tried when all 

 the co-efficients of the polynome are completed. 



2 E 2 In 



